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Show Practical Application of Large Eddy Simulation to Accessible Combustion Chemistry Physics Benjamin Isaa.ca,b,l, Sean Smithb , Alessandro Parente\ Jeremy Thornock b, Philip Smith b ClService d'Aero-Thermo-Mecanique, Universite Libre de Bruxelles, Bruxelles, Belgium b Department of Chemical Engineering, University oJ Utah, Salt Lake City, UT, 84112, USA Abstract Combustion regimes approaching the unity Damkohler number present challenges due to the increase in turbulent intensity and the slowing of reacting time scales due to chemistry dilution and temperature decrease. The physics of these particular systems of interest provide interesting dynamics in which classic models, such as fiamelets, can no longer adequately describe the complex interactions of the system. Many combustion technologies using flameless combustion are emerging and the demand for adequate modeling techniques of these regimes, generally characterized by the unity Damkohler number, are increasing. The chemistry in these systems contains slower reaction time scales which can be accounted for by Large Eddy Simulation (LES) computations. In order to physically define the chemical scales which are accessible on the LES grid, we present a novel approach for the Damkohler definition. FoxFox [7] introduced a method for calculation of the reacting time scales of a system through the eigenvalue decomposition of the source term Jacobian matrix. As a continuation of this work we demonstrate the ability to find the meaningful reacting time scales by performing the decomposition on a red uced set of principal species determined by Principal Variables Analysis (PVA). Through this new methodology we conclude with a demonstration of an a priori analysis which can be done during model selection or model development which clearly defines the accessible physics when using Large Eddy Simulation by providing from the analysis the Damkohler numbers accessible on the LES grid for a given reaction time. 1. Introduction Interaction between mixing and chemical kinetics is the key aspect, which determines the combustion regime. Therefore, a fundamental understanding of turbulence/chemistry interactions in reacting systems may provide the needed "Corresponding author. Phone + 32 2 650 26 80 Fax +32 2 650 27 10 Address: Avenue F. D. Roosevelt 50, 1050 Bruxelles, Belgium. Preprint submitted to Elsevier September 6, 2012 insight into the physics of the flame, allowing an appropriate choice or development of physical models. The interaction between mixing and chemistry may be evaluated through the Damk6hler number (Da), which represents the flow to chemical time-scale ratio. Large Da val ues indicate mixing controlled flames. This is the case, for instance, of the "thin reaction zone" regime, where the flame preserves its laminar structure but it is distorted and wrinkled by turbulent motion. On the other hand, low Da values correspond to slow chemical reactions: reactants and products are quickly mixed by turbulence so the system behaves like a perfect stirred reactor. Flameless (or MILD) combustion is characterized by a strong coupling between turbulence and chemistry, because of slower reaction rates (due to the dilution of reactants) with respect to conventional combustion Cavaliere [3]. It is a generalized opinion that for such combustion regime the Damk6hler number approaches unity Galletti et al. [9]. Indeed lots of modelling investigations have shown that infinitely fast chemistry approaches are not suited. Encouraging results have been obtained through the Eddy Dissipation Concept (EDC) [13] combustion model, especially for its capability of handling detailed kinetic schemes [15]. However, some discrepancies are still observed when using EDC and model modifications have been proposed in the literature for better capturing flameless conditions [1, 6]. The calculation of the Damkohler number requires the definition of appropriate flow and a chemical time-scales. For turbulent conditions, the flow time-scale can be evaluated as the integral time-scale, even though in literature other mixing scales are used (e.g. Taylor scales, Kolmogorov). For instance, in premixed turbulent flames, Veynante and Vervisch [17] suggested to use the integral turbulent time-scale as a mixing time and to estimate the chemical time-scale as the flame thickness to the laminar flame speed ratio. Chemical time-scale calculation poses some issues. In literature several examples of Da calculation are reported, but in most cases just a single global chemical reaction is taken into account to describe oxidation process [3]. For instance, Kuo [12] proposes the following approach for the Damkohler number of a turbulent flame: Da ~ (v~;) (1) where v is the kinematic viscosity, E is the dissipation of turbulent kinetic energy, and Kr is the kinetic constant of the global reaction. On the other hand, Fox [7] provides a method for considering more complex kinetic schemes, suggesting that the chemical time-scale can be defined in terms of the eigenvalues of the N xN Jacobian matrix J of the chemical source terms, whose elements Jij are given by (for an isothermal case): oRi Jij ~ oY J Chemical time-scales can be then associated to each eigenval ues as: 1 Ta ~ [Aa[ (2) (3) where Aa is the eigenvalue vector from the eigenvalue decomposition of J. In a complex kinetic scheme, for which the time-scales can range over several or- 2 ders of magnitude, the slowest chemical time-scale should be chosen for the estimation of the Damkohler number: (4) Such an approach wa.s recently applied by Rehm et al. [16]' who calculated the Damkohler number for a gasification system using the most abundant species (i.e. N ~ 5) to define the Jacobian matrix. Retaining all the species of the kinetic mechanism may, in fact, lead to the determination of non-meaningful time-scales, due to the complete inactivity of some species in specific region of the flame. However, the choice of the species to be retained is not to date established and it is generally performed arbitrarily a.s in [16] . The first of two objectives for the present paper is that of defining a methodology for the determination of the principal variables of a reacting system, to allow the determination of the involved chemical time-scales from complex reaction schemes. The Damkohler number's usefulness can be extended to large eddy simulation by making a relationship between the scalar energy spectra and the Damkohler number. If one could clearly define the resolvable Damkohler numbers on an LES grid it would be very simple to choose appropriate models and or develop models which exploit most of the information made available by the resolved part of a LES simulation. For this purpose, the current paper outlines an a priori method for estimating the scalar energy spectra Fox [7] and the relation between the Damkohler number and the wave number. 2. Methodology Principal Component Analysis [11, 10] is a statistical technique employed in the analysis of multivariate data-sets, for detecting the directions that carry most of the data variability, thus providing an optimal low-dimensional projection of the system. For a data set, X, consisting of n observations of p variables, the sample covariance matrix, S, of X can be defined a.s S ~ 1/ (n - 1) XTX. Recalling the eigenvector decomposition of a symmetric, non singular matrix, S can be decomposed a.s S ~ ALAT, where A is the (Q x Q) matrix whose columns are the eigenvectors of S, and S is a (Q x Q) diagonal matrix containing the eigenval ues of S in descending order, h > b > ... > lp. The covariance matrix indicates, therefore, the level of correlation between the non-dimensional variables. Values close to zero denote uncorrelated variables whereas correlations close to one indicate strongly correlated variables. Based on the correlation values, the redundant and unuseful information contained in the original datasets can be easily removed. Once the decomposition of the covariance matrix is performed, the Principal Components (PC), Z, are defined by the projection of the original data onto the eigenvectors, A, of the covariance matrix, S, Z = XA. Then, the original variables can be stated as a function of the PC as X = ZA T, being A orthonormal and, hence, A-I = AT. Nevertheless, the main objective of PCA is to replace the p elements of X with a much smaller number, q, of PC, preserving at the same time the amount of information originally contained 3 in the data. If a subset of size q « Q is used, the truncated subset of PC 1S Zq = XAq. This relation can be inverted to obtain: (5 ) where Aq is the matrix obtained by retaining only the first q col umns of A. The linear transformation provided by Eq. (5) ensures that the new coordinate axes identified by PCA are orthogonal and they provide independent and decreasing contributions to the amount of original variance explained by the PC. Thus, if only the subset Aq of A is retained, Xq represents the best q-dimensional approximation of X in terms of squared prediction error. Variables are generally centered before PCA is carried out, to convert observations to fluctuations on the mean; moreover, scaling is applied when the elements of X have different units or when they have different variances, as it is the case for this investigation. A centered and scaled variable can be defined as: x· -x· Xj = J J dj (6 ) where dj is the scaling parameter adopted for variable Xj. Several scaling options are available, incl uding normalization by the variable range, standard deviation, maximum and average values. The present paper uses the standard deviation as scaling parameter. This ensures that all the elements of the scaled X matrix have a standard deviation equal to one, giving them similar relevance. 2.1. Principal Variables Principal variables (PV) represent an attempt to help the physical understanding of Principal Components. PV algorithms try to link the PC back to a subset of the original variables, which satisfies one or more optimal properties of PCA. A number of methods exist for selecting a subset of q original variables which preserve most of the variation in X. The method considered in the present analysis is called B2 backward method [11[. In the B2 method, PCA is performed on the original matrix of Q variables and n observations. The eigenvalues of the covariance/correlation matrix are then computed and a criterion is chosen to retain q of them. This will lead to discarding Q - q variables, which are eval uated starting from the last component, looking for the variables corresponding to highest eigenvector coefficient. Those variables are then discarded, as they are highly correlated with a component not carrying any useful information. The variables extracted with the principal variable algorithms are used to compute a subset of the full Jacobian matrix, only including the derivatives with respect to the selected principal variables. This allows the determination of the modes to be compared with the ones provided by the full Jacobian matrix, incl uding all the species involved in the original detailed kinetic mechanism. In all cases, the determination of the Jacobian matrix is performed with the MATLAB@ code JACOBEN. The code is particularly interesting as it formulates the chemical source term equations as symbolic expressions and then uses the 4 symbolic differentiation function in MATLAB@ to form the analytical expressions for the derivatives of the chemical source terms with respect to chemical species. The code requires the chemical mechanism to be in CHEMKIN format, as well as all thermodynamic state space parameters describing the turbulent combustion, including the species mass fractions and temperature. A Jacobian matrix is then evaluated for every grid point provided in the thermodynamic state space input file. 2.2. Energy Spectm After computing meaningful chemical time scales using the proposed method, it is possible to develop an a priori tool which can relate the scalar energy spectra to the Damkohler number, allowing for a definition of resolved Damkohler numbers in the context of LES simulations. The scalar energy spectra as outlined by Fox Fox [7] gives the following definition for the scalar energy spectra: (7) where Coe Rj 2/3 is the Obukhov-Corrsin constant, E¢ is the scalar dissipation rate, E is the turbulent dissipation rate, v is the viscosity of the mixture, f'i, is the wavenumber, 77 is the kolmogorov scale 77 = (VE;8) 1/4 . The scaling exponent 13 is given by: (8) The non-dimensiona;:~:L~~v~ c(utOff fu::~ons ~~~ )gi:/e3r:~y: [("Lu )2 + CL] (g) fry (M/) ~ exp [-13 U (m/)4 + c~l'/4 - Cry) ] (10) here the turbulent length scale is Lu = k~2, the diffusive-scale cutoff function is give by: fD (M/) ~ [1 + CD Sc-d(,ry)/2 "'I] exp [-cDSC-d(,ry)/2 "'I] (11) where CD = 2.59. Diffusive exponent is given by: 1 1 d ("'I) ~ 2: + 4fry ("'I) (12) The Batchelor-scale cutt off function given by Kraichnan is given by: fB ("'I) ~ [1 + CdSc-d(,ry) "'I] exp [-cDSc-2d('ry) ("'1)2] (13) The scalar dissipation constant Cd is calculated given the follow constraint: = ! ("'1)2-~('ry) h ("Lu ) fB("'1)d ("'I) ~ eSc 2 00 o CL and Cry are constrained by the following equations: 5 (14) = ! 2v,,2 Eu (,,) ~ E o (15) (16) The scalar energy spectra could be calculated at each point in time at every grid point in a domain. In order to solve the equations presented here one would need to provide definitions for the following variables: density (p), viscosity (v), turbulent kinetic energy (k), energy dissipation rate (E), mixture diffusion coefficient (f), and scalar variance ((1,12)). One can directly relate the Damk6hler analysis to LES by finding a relationship for wavelength to a mixing time scale. In order to do this the following definitions are required: • scalar mixing time - which is the integral time scale of the scalar mixing (17) • scalar energy dissipation rate - which is the the rate at which the scalar energy is dissipated (18) • the scalar variance - related to the mixture fraction variance = (1,12) ~ ! E¢d" (19) o Next one can simply derive an expression for scalar mixing time as a function of the wave number: (20) 3. Test Cases First a demonstration of the Principal Variable approach to the Damkohler number is given based on a flameless combustion data set where a unity Damkohler number is expected. The data used in the present work refers to the jet in hot co-flow (JHC) burner designed by 151 to emulate flameless combustion conditions. It consists of a jet of a CH4/H2 mixture (inner diameter D = 4.25 mm) within an annulus co-flow (inner diameter ID = 82 mm) of hot oxidizer gases from a porous bed burner mounted upstream of the exit plane. The entire burner is placed inside a wind tunnel feeding air at the same velocity as the hot 6 co-flow. The datasets used in the present work refer to a jet Reynolds number of around 10,000 and different oxygen ma.ss fraction, i.e. 3% (HMl), 6% (HM2) and 9% (HM3) in the co-flow. The jet Reynolds number is around 10,000 for all flames. The available data consists of the mean and root mean square (RMS) of temperature and concentration of major (CH4 , H2" H20, CO2 , N2 , and O2 ) and minor species (NO, CO, and OH) at different locations. A detailed description of the systems and tests can be found in [4[. The JHC wa.s modeled with the Fl uent 6.3 software by Ansys Inc. A 2D axisymmetric domain was chosen with a structured grid made of 25k cells. The steady-state Reynolds-Averaged Navier-Stokes (RANS) equations were solved with a modified version of the k-e turbulence model (i.e. imposing Cd ~ 1.6 for round jets [14[. The KEE-58 mechanism [2] consisting of 17 species and 58 reversible reactions was employed for the oxidation process. Turbulence/chemistry interactions were modeled with the EDC model. The constant of the residence time in the fine structures was set to 1.5 as this was found to improve substantially the predictions of temperature and chemical species in flameless conditions [1, 6]. For the boundary conditions, velocity inlet conditions were given to the unmixed fuel jet, co-flow oxidizer and tunnel air, paying particular attention to turbulent intensity [1]. The discretization was made with a second-order upwind scheme, whereas the pressure-velocity coupling was handled with the SIMPLE algorithm. Residuals for all equations were kept lower than 10-6 as a convergence criterion. The temperature and CO mass fraction were also monitored at the exit plane as another convergence criterion. . The numerical simulation results obtained from the simulation of the JHC burner have been successfully validated [1 [ against the available experimental data and they represent an ideal data-set for testing the proposed methodology for chemical time-scale calculation in case of complex kinetic mechanisms. In the following, the effect of the principal variables on the chemical time-scale determination is addressed. In particular, emphasis is put on the minimum number of variables required to well capture the reaction region of the flame. 4. Results and Discussion Figure 4 shows, from top to bottom, the temperature T, the chemical timescale, Te, and the Damkohler number, Da, as a function of mixture fraction, e. The data are plotted along the burner axis, for the systems HMI-HM3 using the full Jacobian matrix. From the analysis of the time-scale and Da values it is clear that keeping all the variables in the Jacobian matrix does not help identifying the relevant processes for the system under investigation. In particular, if no filtering is applied to the original thermo-chemical state variables, the analysis will point out the existence of extremely slow time-scales of the non-reacting species, as shown in Figure 4, leading to Da val ues close to zero, for all three systems. A first attempt is then that of filtering out from the Jacobian analysis all the scales, i.e. all values above 1000 (limit for slow chemistry processes according 7 ~:th-:::=::=:=~ 'OWoo o.()( 0.05 0.08 0.07 0.08 009 Hguro 1: Flcrn 4 to botu:tn: tompor 'tuu, T, m.rnic,,) tirn. "'.J., T., md DmtkOOl", wlu", '" a functictl ci mi:rt.ur. fncti ctl ,~. Full hcobi.1Ul mm "" =d unfijw od tim. "">.!,,, \0 fill. Thill r",ults in t he plot:; ~hown in Figure 4, ~howing the t im,,"~calElS and Da value~ for filtered Jacobian matrix. Howeve r, al:lo in thi:; case, t he t ime-~ cale analy:;i:; do ,.; not provide a clear ind g ht int o t he invest ig ated combu15tion ~ystem. T he Do. valuas obtai ned for t he t hree Ca:las appear very clooe and ~imply ~hifted along the mixture fract ion ;ail; going from ca:;e HM l t o ca:;e HM3. More important ly, t he Da value~ are in all case~ fairly large C H5), thu:; far from what would be exp€cted in flamel"", condition:;. T he rellult:; ~hown in Figure 4 indicate t hat keeping all t he t hermo-chemical ,;tate variablas in t he Jacobian mat rix do e~ not allow identifYing t he controlling chemical time-"" a le of t he :sy:;tem. It i~ t herefore vel)' important to identifY t he relevant variable:; for the t im e--~cale analy:;i:; t hro ugh a rig orou~ selection method. Figure 4 ~how:; in black and red, re:;pectively, t he chemical t ime--~c ale~ =0- dated to t he J acobian mat rix oby:;tem HMl wit h all t he ~t ate variable~ and with a ~ing le PV, determined wit h met hod B2 (S ection 2.2). It can be oooerved that t he ~ing le PV, CO, in the present case, allow:; capt uring the d ow dynamic:; of t he react ing :sy:;tem without ~howing the peak:; di:;played by the large", (meaningful) time-~cale of the full ~y:;tem, determined by the int eract ion:; be-twe en the different chemical ~pede~. The limiting t im e--~cale~ obtained in t he two c ase~ overlap fairly well form ~ > O,O ~. However, below t hat value, t he two curve~ ~how major di:;crepande~, t he black dot~ being about four orde"" of magnit ude below the red one~. Thi~ difference i:; respon~ible for t he large Da value:; ~hown in Figure 4. Figure 4 ~hows t he chemical time-~cale, T" and Damkohler, Do., value~ as a :;;; :::r ;H ; .; : • • 1 0 " \°0.02 ,ro ,~ ,re ,re ,~ ". '''' " • ::l~ • c I· :~I 900 0.00 0.0.1 0._05 0._06 ,~ , '''' '''' " Hguro:c Flcrn 4 to botu:tn: tompor 'tuu, T, m.rnic,,) tirn."'.J., T., md DmtkOOl", wlu", '" a functictl ci mi:rt.uro f,..:;tictl ,~. full J ",obim rum"". Tim."">.!,, aoo..;, 1000 filtor~ M _. 8Ia<O . F .. SysIOM , _ . p • • _ 10'1r-----__ ----__ ----______ .----__________ ----, , "'gur< 3: Clwruic.J """,,,,,>.los ~.ud to th. full (tirl) md r~ocod (rod) J",obiID mat<!" (rod Un.) Rodocod J",obim o"MinM '-"'ng ctl. PV, drumninM usffig m.th:xI B2 HMl c,"", F>guu 4: Flcrn 4 to botu:tn: m.rnicol tim~>.l., ' 0, md DlIDlloOh1or wlu., "" ~ functictl ci mi:rt.ur. f''''ti<:tl, (. J ",obim m m i" r= ict.<d 0". F<incipol =i~"" function of mixture fraction,~, for t he Jacobian mat rix calculat ed udng a ~ingle PV, i. e. CO" and for all flamas. Differently from Figure 4, the rewlt:; indicate for t he pre~e nt ca:;e a meaning ful t rend, ~howing Do. number:; around unity for the HM1 ca:;e, wppooed to be reprasentative of a fl amele",,; combust ion regime. The Damkohler valuas increa:;e and almoot double when going from HM1 t o HM3, i.e. increa:;ing t he oxygen in the co-flow from 3% to cjf'o and moving, t here fore, t o convent ional flame condition:;. Very int erastingly, t he Do. valuas do not change when incre a:;ing t he number of principal variable~ a:; long a:; the number of PV ill kept below 12. At thill point, the analy:si~ of the time,,;calas ~how:; again the appearance of large chemical t im"", which are relat ed t o inactive t hermo-chemical variable~ and which ~hould be not co n:;idered in t he analy:;iIl. Thill ill confirmed by the analy:;iIl of Fig ure 4, which ~hows t he Do. value~ a:; a function of ~ along the burner ;JJ:iIl. When the number of PV equal:; 12, the Da value~ drop to zero (green dot:;) due to the appea rance of large chemical time-~calas. The propooed methodol ogy ~ee= to provide a very robust way for the detenninat ion of t he limit ing tim,,"~cale =ociated t o a chemically react ing ~y:;te m. In particular, it can provide the variable~ that ~hould not be included in a time-"" a le ana ly:;iIl. An example of a ~calar ene rgy ~p ect ra a:; well a:; t he Energy "Pectra ill ~hown in Figure 6. AI; the Schmidt number increa:;"" t he mixture diffusivity increa:;"" leading to a slower dissip ation of energy. In contras/;, a:; the mixt ure diffusion increa:;"" t he ~c alar energy dissipation occur:; ra pidly. The ""alar energy ~pect ra ill commonly used in de~cribing t he concept ofLES, where it ill de~ire d to re"",lve 80% of the energy ~pectra, rasolving t he large energy cont a ining ~cale~ of t he ~y:;tem. , ~,. , D>rnI<&I .. ;o1,... .. ~ !unc"",,, OftM mixtu,. f""tm .Jon< tho. """'" .m fa. fuza. HM1, "";t", tl.. am"" of >ri",ipo.! _ .u.. Full j",otun m>tr'" Iil","; to '''''''''' ",>1 .. ,,..,,.,, thu. 1(0) • n '" - Sc=O.OOOl - Sc:{).Q1 " - SC=l - SC=l00 " - SC= l000(1 ~ 1 >' r ; , -, "-:< - 1 0 ~ -" -'" - 25 0 0.' " .. , Figure 7: Scalar mixing time as a function of wavenumber Figure 7 shows the scalar mixing time as a function of the wavenumber for various Schmidt numbers. Given reaction times from the Jacobian time scale analysis presented in this paper one can now calculate the Damkohler number as a function of wavenumber. This allows now for a definition of Damkohler numbers which can be fully resolved in the LES framework and Damkohler numbers which will need to be modeled for a given chemistry system. Figure 4 shows an example of the a priori LES Damkohler analysis for the H1vIl case. The section in yellow can be resolved by LES, the section in gray represents the Damkohler numbers of the reaction, and the intersection of these areas indicate the Damkohler numbers which can be resolved, The colored lines represent time scales ranging from 2.3e - 07 s to 0.003 s as calculated from the time scale analysis outlined in the paper. Such an analysis indicates the ability of a combustion model in a specific combustion regime. Figure 4shows the weights for the eigenvectors corresponding to the calculated time scales. These pie charts show the sensitity of the chemical time mode to the principal variables. As would be expected the slow time scales correspond to species such as CO2 , O2 , N2 and rapid time scales correspond to radical species where reaction times are expected to be very small. 5. Conclusions A proposed methodology is suggested for the calculation of Da in case of complex kinetic schemes. In particular, Principle Component Analysis and Principal Variables based approaches appear well suited to the purpose, as they are able to identify the directions and variables carrying most of the information 12 , log,,) y.) 111m) Figure 8: Resolvable Damkohler numbers for a given reaction time in LES system in a multivariate system. The methodology was successfully tested on a flameless combustion data sets, for which validated numerical data obtained using a complex kinetic scheme are available. In particular, the proposed methodology was able to identify the limiting time-scale for different configurations, yielding to a subset of variables (called principal variables) for which the Damk6hler resulted invariant. In addition an a priori study was performed to identify the Da numbers which can be actually resolved in a LES computation, allowing the development of appropriate combustion models, targeted to specific applications. Acknowledgments Part of the present research was sponsored by the National Nuclear Security Administration under the Accelerating Development of Retrofittable C02 Capture Technologies through Predictivity program through DOE Cooperative Agreement DE-NA0000740". References [1[ J. Aminian, C. Galletti, S. Shahhosseini, and L. Tognotti. Key modeling issues in prediction of minor species in diluted-preheated combus-tion conditions. Applied Thermal Engineering, 31(16):3287 - 3300, 2011. ISSN 1359-4311. doi: 1O.1016/j.applthermaleng.2011.06.007. URL http://www.sciencedirect.com/science/article/pii/S1359431111003176. 13 GTS . 5.157B<i_DJ GTS . O.OO'J27227 GTS . O.003151' GTS . Q.353 1o_DJ C02 100% H84% G TS • 1.70050-D€ GTS.2.M80_DJ GTS.2.6€390_DJ GTS.=W 037% Figure 9: Resolvable Damkohler numbers for a given reaction time in LES system 14 [2[ R.W. Bilger, S.H. Starner, and R.J. Kee. On reduced mechanisms for methane-air combustion in nonpremixed flames. 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